It is believed that most actively managed funds fail to outperform the overall market. To empirically test this statement, an analyst has collected the following secondary data to examine whether or not the mean return of an index which represents all the actively managed funds in the market is statistically different from the average return of the overall market.
Indexes # Observations Average Annualized
Fund 101 12.10% 15.73%
Market 101 4.60% 9.98%
a. Conduct an appropriate test to examine whether the two samples have the same population variances at the 2% level of significance.
b. Assume that the two samples have the same variances; conduct an appropriate test to examine whether the two indexes have the same mean at the level of 0.01.
c. Conduct appropriate test to examine whether the annualized mean return for the active fund index is different from 9% at the 0.01 significance level.
d. Construct a 95% confidence interval for the mean monthly returns of benchmark market index.
e. Compare and contrast the use of test statistic/critical value approach and the p-value approach to condut a hypothesis test.
a. F = s1^22/s2^22
F = 0.15372/0.09982 = 2.372
H0: variances are equal
H1: variances are not equal
Since critical value for 2% level of significance 31.821>2.372, we reject H0
Hence variances are not equal
b. t = 12.10 - 4.60/[sqrt(0.1537^22/101 + 0.0998^2/101)]
=0.075/sqrt[(0.0236+0.00996)/101] = 0.075/0.0001804 = 4.11
T value for 0.01 level of significance = 2.626
Since 4.11>2.626 we reject H0 ...
The expert determines whether actively managed funds fail to outperform the overall market.